Monday, January 21, 2013

In January of 2013, about a month after the horrific
shootings of children in Newtown, Connecticut, the Pew Research Center released
a survey of gun-related political leanings of people in America. They first asked the respondents to identify
themselves as either gun rights proponents, or gun control proponents. They then asked the respondents questions about their
political activity: did they contribute money to organizations that took a
position on gun policy? Had they
contacted a public official to express an opinion on gun policy? Had they signed a petition on gun
policy? Etc. The results indicated that those who
prioritized gun rights were 1.7 times more likely to have been politically
active (i.e., participated in one or more of these activities) than those who
prioritized gun control. Why should gun
rights advocates be almost twice as likely to be politically active than gun
control advocates?

To understand this behavior, it is useful to consider how
the human brain makes choices when faced with gains and losses.

In 1990, Kahneman and colleagues performed an experiment in
which they selected some participants and gave them a coffee mug as a gift. They then
asked them to assign a minimum price on the mug that they were willing to sell
it. These participants asked for about
$7. They then took another group of
participants and showed them the same mug and asked how much they would be
willing to pay to own it. They responded around $3. Knetsch (1989)
found that people who are given a chocolate bar want $1.83 to sell it, but will
pay only $0.90 to buy it. The difference
in the two prices is explained by loss aversion: the sellers evaluate the choice
of giving up something that they already own by viewing it as a psychological loss. In order to compensate for that loss, they request a lot of money. Buyers, on the other hand, evaluate the choice as a psychological
gain. They are willing to pay much less for the pleasure that they perceive in owning it.

In general, the pleasure that you feel if someone was to
give you an item tends to be much less than the pain you feel if you were to own
that item and were to lose it. This is
called an endowment effect.

Carmon and Ariely (2000) explain this behavior by suggesting
that when faced with loss of something (e.g., selling), people focus on their
sentiment toward surrendering the item (and not the money that they are
gaining), whereas when faced with gain of something (e.g., buying), people
focus on their sentiment toward what they forgo (typically money, and not the
item they are gaining).

Now let us consider the question of why gun rights
proponents are more politically active than gun control proponents. The current political climate is one in which
the President and the Congress are considering laws that would limit gun
rights. This is viewed as a loss to gun
rights proponents. In contrast, the same
laws are viewed as a gain for gun control advocates.

The gun rights proponents (but not the gun control proponents) are under the influence of the endowment effect because
if the proposed laws are enacted, it would result in a loss of what they already
‘own’. For them, the proposed laws carry
a negative psychological value. If we could generalize
from behavioral economics literature, we would speculate that this negative value is about twice as
large as the positive psychological value that would be gained from the
perspective of gun control proponents. This
may be the reason why the gun rights proponents are about twice as likely to be
politically active as the gun control proponents.

The deeper idea is that any change from the status quo will meet
with much stronger resistance by those who view the change as a loss, as
compared to the enthusiasm that it fosters in those who view the change as a
gain.

References

Carmon, Z. and Ariely, D. (2000) Focusing on the forgone:
How value can appear so different to buyers and sellers. Journal
of Consumer Research 30:15-29.

Kahneman D., Knetsch J., and Thaler R. (1990) Experimental tests
of the endowment effect and the coase theorem.
Journal of Political Economy
98:1325-1348.

Thursday, January 10, 2013

How do we know when a data point is an outlier? Take a look at the figure below. It represents 15 data points that were
gathered in some experiment. Would you
say that the left-most point is an outlier?

Maybe the instrument that collected this data point had a
malfunction, or maybe the subject that produced that data did not follow the
instructions. If we have no other
information than the data, how would we decide?

When we say a data point is an outlier, we are saying that
it is unlikely that it was generated by the same process that generated the
rest of our data. For example, if we
assume that our data was generated by a random process with a Gaussian
distribution, then there is only a 0.13% chance that we would collect a data
point that is 3 standard deviations from the mean. So what we need to do is try to estimate the
standard deviation of the underlying process that generated the data. Here I will review two approaches, and then
show how successful they are in labeling outliers.

Median Absolute Deviation (MAD)

Hampel (1974) suggested that we begin with finding the
median of the data set.

Next, we make a new data set consisting of the distance
(this is a positive number) between each data point and the median. Finally, we find the median of the new data
set. That is, we compute the following:

MAD = b median( abs(x – median(x) ) )

If we set b=1.4826, then MAD is an estimate of the standard
deviation of our data set, assuming that the true underlying data came from a
Gaussian distribution. For our data set
above, here is the estimate of the standard deviation, centered on the median:

Based on MAD estimate of the standard deviation, we would
say that the left-most data point is indeed more than 3 estimated standard
deviations (MADs) from our estimate of the mean (the median).

So a typical approach is to label as ‘outlier’ a data point
that is farther than 3 times the MAD (standard deviation) than the median of
the data. That is, compute the following
for each data point:

abs(x – median(x) ) / MAD

Label as ‘outlier’ the data points for which this measure
gives you a number greater than 3. But
how good is this method? To check it, I
did the following experiment. I generated data sets drawn from a normal distribution with a
constant mean and standard deviation, and then computed the probability of a
false positive, that is, I computed how likely it was that a point would be
labeled as outlier by MAD, when in fact it was less than 3 standard deviations
from the mean. Here is the resulting
probability, plotted as a function of the data size:

The above plot shows that when the data set is small (say 10
data points), about 20% of the data points that the algorithm picks as outliers
are in fact within 3 standard deviations of the mean. As the data set grows larger, the probability of
false positives declines and the algorithm does better. But even for a data set of size 20, there is
better than 15% chance that the bad data point is in fact not bad.

Median Deviation of the Medians (MDM)

Rousseeuw and Croux (1993) suggested a method that, as we
will see, is better. For each data point
xi, we find the distance to all other data points and find the resulting
median. We do this for all data points
and we get n medians. Now we find the
median of this new data set:

MDM = c median( median( abs(xi –xj) ) )

If we set c=1.1926, then MDM is a robust estimate of the
standard deviation of the data set, assuming that the true underlying data came
from a Gaussian distribution. For our
data set above, here is the estimate of the standard deviation:

To check how this method compares with MAD, I generated
data sets drawn from a normal distribution with a constant mean and standard
deviation, and then computed the probability of a false positive, that is, I
computed how likely it was that a point would be labeled as outlier by MDM,
when in fact it was less than 3 standard deviations from the mean. Here is the resulting probability, plotted as
a function of the data size:

The
above plot shows that regardless of the size of the data (here ranging from 6
data points to 20), a data point that MDM labels as an outlier has about 9%
chance of being a false positive, i.e., not an outlier. For small data sets, MDM is two to three
times better than MAD.

References

Hampel FR (1974) The influence
curve and its role in robust estimation. Journal of American Statistical
Association 69:383-393.

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About Me

I was born in Iran and immigrated to the US at the age of 14. I was educated at Gonzaga University, University of Southern California, and finally MIT. I studied under the mentorship of Prof. Michael Arbib and Prof. Emilio Bizzi. I am currently Professor of Biomedical Engineering and Neuroscience, and the Director of the BME PhD Program at Johns Hopkins School of Medicine. I am a neuroscientist who uses mathematics to understand how the brain controls our movements.